Describing Transformations of Quadratic Functions

We want to match the given function with its graph. We will begin by noticing that the given function is a transformation of the form $y=a(x-{\color{#0000FF}{h}})^2+{\color{#009600}{k}},$ where ${\color{#0000FF}{h}}$ represents a horizontal translation and ${\color{#009600}{k}}$ represents a vertical translation. $\begin{gathered} y=x^2+{\color{#009600}{1}} \end{gathered}$ In our case, we have ${\color{#009600}{k}}={\color{#009600}{1}}.$ This means that the graph of $y=x^2+1$ is obtained by translating the graph of $y=x^2$ up by ${\color{#009600}{1}}$ unit.

Therefore, the correct answer is option C.